Conditional Probability on $\Sigma$-Complete Boolean Algebras

Abstract

Probability as measure on a Boolean algebra was presented by Kappos [5], but a treatment of conditional probability relative to a subalgebra is missing. The Stone space of a $\sigma$-complete Boolean algebra (see [10], p. 24) enables one to apply the concepts of conditional probability for a $\sigma$-algebra of subsets of some space (see [2], pp. 15-28), but the problem deserves closer attention. In this note we consider conditional probability with respect to a $\sigma$-subfield of the $\sigma$-field generated by the open-closed subsets of the Stone space of a Boolean $\sigma$-algebra. We show that there is always a regular conditional probability (see [4], p. 80) relative to a full $\sigma$-subalgebra of Baire sets. With a modified definition of probability on a Boolean algebra a treatment of conditional probability is possible without reference to the Stone space. For this a generalized integral is defined and the theory of integration is begun for it. A definition of conditional probability on a $\sigma$-complete Boolean algebra is given for which there is no regularity condition. We conclude the discussion with a study of the relationship of this theory with the conventional theory.